Obstructions for bounded shrub-depth and rank-depth

Abstract

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný, Kwon, Obdržálek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.

19 pages, 5 figures; accepted to Journal of Combinatorial Theory Ser. B

References in corpus (5)

• The grid theorem for vertex-minors
• Branch-depth: Generalizing tree-depth of graphs
• Classes of graphs with no long cycle as a vertex-minor are polynomially $\chi$-bounded
• Scattered classes of graphs
• Graphs of bounded depth-$2$ rank-brittleness

Cited by in corpus (3)

• Branch-depth: Generalizing tree-depth of graphs
• Graphs of bounded depth-$2$ rank-brittleness
• Some classical model theoretic aspects of bounded shrub-depth classes